This note mainly serves as an introduction to the techniques used in the papers [1]}, [2]},
which are concerned with the local decay of waves on a Schwarzschild background. The decay estimates are obtained by separation of variables and
the analysis of the flow for each angular momentum in [1]}. By means of a semiclassical WKB analysis involving a global Liouville-Green transform, as well as semiclassical Mourre theory at energies near the top of the barrier, [2]}
sums up these fibered estimates
over all angular momenta incurring the loss of finitely many angular derivatives.
Note that [1]}, [2]} are not entirely self-contained and rely in part on [7]}, [8]},
[9]}, [10]}, [11]}. As shown in these references, the Schrödinger flow can be analyzed analogously. The original motivation for [7]}, [8]}
was to study the long-term dispersive behavior of solutions to
Schrödinger and wave equations on specific non-compact manifolds exhibiting closed geodesics, such as the hyperboloid of one sheet.
In analogy with the unique periodic geodesic on such a hyperboloid, which is exponentially unstable, the surface of closed geodesics
around a Schwarzschild black hole is known as photon sphere and corresponds to the collection of all periodic light rays. The photon sphere is also unstable.
